Proofs from THE BOOK

Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book."^[1]

Proofs from THE BOOK

Authors	Martin Aigner, Günter M. Ziegler
Illustrator	K. H. Hofmann
Language	English
Subject	Mathematical proofs
Publisher	Springer
Publication date	1998
Pages	239
ISBN	3-540-63698-6

Content

Proofs from THE BOOK contains 32 sections (45 in the sixth edition), each devoted to one theorem but often containing multiple proofs and related results. It spans a broad range of mathematical fields: number theory, geometry, analysis, combinatorics and graph theory. Erdős himself made many suggestions for the book, but died before its publication. The book is illustrated by Karl Heinrich Hofmann [de]. It has gone through six editions in English, and has been translated into Persian, French, German, Hungarian, Italian, Japanese, Chinese, Polish, Portuguese, Korean, Turkish, Russian, Spanish and Greek.

The American Mathematical Society awarded the 2018 Leroy P. Steele Prize for Mathematical Exposition to Aigner and Ziegler for this book.

The proofs include:

• Six proofs of the infinitude of the primes, including Euclid's and Furstenberg's
• Proof of Bertrand's postulate
• Fermat's theorem on sums of two squares
• Two proofs of the Law of quadratic reciprocity
• Proof of Wedderburn's little theorem asserting that every finite division ring is a field
• Hadamard's maximal determinant problem and Hadamard's inequality
• Four proofs of the Basel problem
• Proof that e is irrational (also showing the irrationality of certain related numbers)
• Hilbert's third problem
• Sylvester–Gallai theorem and De Bruijn–Erdős theorem
• Cauchy's theorem
• Borsuk's conjecture
• Schröder–Bernstein theorem
• Wetzel's problem on families of analytic functions with few distinct values
• The fundamental theorem of algebra
• Monsky's theorem (4th edition)
• Van der Waerden's conjecture (6th edition)
• Littlewood–Offord lemma
• Buffon's needle problem
• Pigeonhole principle and double counting, Sperner's lemma
• Sperner's theorem, Erdős–Ko–Rado theorem and Hall's theorem
• Lindström–Gessel–Viennot lemma and the Cauchy–Binet formula
• Four proofs of Cayley's formula
• Kakeya sets in vector spaces over finite fields
• Bregman–Minc inequality
• Dinitz problem
• Steve Fisk's proof of the art gallery theorem
• Five proofs of Turán's theorem
• Shannon capacity and Lovász number
• Chromatic number of Kneser graphs
• Friendship theorem
• Some proofs using the probabilistic method